1 Data sequences and cumulative distribution functions

Question 1

(D) CDF (3)

Answer 1

F : R → [0, 1] is called cumulative distribution function if

1. lim_t→-∞ F(t) = 0, lim_t→∞ F(t) = 1
2. F monotonically increasing
3. F right continuous

Question 2

(D) ECDF

Answer 2

x1, …, xn data sequence. Then the ECDF of the data sequence is defined by

Fn : R → [0, 1], Fn(t) := hn((−∞,t]) = 1/n ∑ 1_(−∞,t] (x_i).

Question 3

(A) Distribution of FX(X)

Answer 3

F stetig, X hat CDF F. Dann gilt

F(X) ist gleichverteilt auf (0, 1).